Which way does train 8710 go when heading east from Jamaica? Does it go via the main line/Hicksville, or does it go via the Babylon line? I'm just curious to find out which way it will go to find out what I'll see when I go for a ride at the end of the month.

I haven't taken that specific train myself, so I can't answer definitively, but according to the MTA's GTFS data (which is used in apps like Google Maps or Transit) it's currently scheduled to use the Babylon/Montauk branch. No idea if that was always the case. Not directly related to your question, but I believe sometime last year the MTA rerouted 8700 (the late night weekend Jamaica to Montauk train that used to stop at Hicksville) from the Main Line onto the Babylon/Montauk branch due to prospective construction on the future third track, so I would assume that any and all weekend Montauk Branch trains that normally run/ran on the Main Line and Central Branch could be liable to be rerouted via the Babylon/Montauk branch.

DB and RDE: True-Any Montauk Branch train to east of Babylon that does not stop at eitherMineola or Hicksville can travel by way of either route to Babylon. Trains can make better timeon the electrified segment to Babylon than they could over the Main Line/Central Branch taking note to the Third Track Project work in which trains can encounter delays due to single trackuse on parts of the route during construction work. MACTRAXX

EXPRESS TRAIN TO NEW YORK PENN STATION-NO JAMAICA ON THIS TRAIN-PLEASE STAND CLEAR OF THE CLOSING TRAIN DOORS

Unless if a Montauk Line train has any of the Main Line stops on the schedule, I think the dispatcher probably prefers to have them run by way of the Babylon Line-no grade crossings and constant 80 mph running. Even when the third track on the Main Line is in service, LIRR trains heading to Montauk by way of the Main Line and Central branch would still have to play around with the grade crossings out there. There are many of them.

On any given day it's a 50/50 chance for either route to Babylon unless there's single tracking on the Main line or on the (middle)Montauk branch,which can force a move on the other route to avoid delay.Believe the LIRR will keep both Main lines open during the AM/PM Rush Hours while the 3rd main line is constructed.But on Weekends with trackwork in progress on both routes, the DS/204 will pick the route with the least possible delay(hopefully)